📚 B9i Part 3: Vector Geometry and Proof | B9i 第三部分：向量几何与证明
Vectors are one of the most powerful tools in geometry because they allow you to describe positions, directions and relationships without relying only on coordinates or diagrams. In B9i Part 3, the focus moves from simple vector arithmetic to geometric proof: parallel vectors, collinear points, midpoints, ratios and vector equations of straight lines.
向量是几何中最有力的工具之一，因为它们能够不依赖坐标或图形来描述位置、方向和关系。在 B9i 第三部分，重点从简单的向量运算转向几何证明：平行向量、共线点、中点、比例关系以及直线的向量方程。
1. What B9i Part 3 Actually Tests | B9i 第三部分考查的核心内容
This part usually combines vector operations with geometric reasoning. You are expected to interpret vector expressions such as AB = b − a, use scalar multiples to identify parallel vectors, and justify why points lie on the same straight line. The key skill is writing a clear chain of vector equalities.
本部分通常将向量运算与几何推理结合起来。你需要理解 AB = b − a 等向量表达式，利用数乘关系识别平行向量，并证明点共线。核心技能是写出清晰的向量等式链。
Typical questions ask you to prove that a point is the midpoint of a line segment, that two lines are parallel, or that three given points are collinear. Each proof should finish with a conclusion in ordinary geometric language.
典型问题会要求你证明某点是线段的中点、两条线平行，或三个已知点共线。每个证明最后都要用普通的几何语言写出结论。
2. Vector Notation and Basic Operations | 向量记号与基本运算
A vector has both magnitude and direction. You can write a vector in column form, in bold type, or with an arrow above the letter. Addition follows the triangle rule: AB + BC = AC. Scalar multiplication stretches or reverses a vector but keeps it parallel to the original.
向量既有大小又有方向。你可以用列向量、粗体字母或字母上方加箭头来表示向量。加法遵循三角形法则：AB + BC = AC。数乘会拉伸或反向一个向量，但保持它与原向量平行。
AB + BC = AC
For example, if a = (2, 3) and b = (−1, 4), then a + b = (1, 7). Subtraction is performed componentwise, so a − b = (3, −1).
例如，若 a = (2, 3)，b = (−1, 4)，则 a + b = (1, 7)。减法按分量进行，因此 a − b = (3, −1)。
| Operation 运算 | Rule 规则 | Example 示例 |
| Addition 加法 | Add components 分量相加 | (3, 2) + (−1, 5) = (2, 7) |
| Scalar multiplication 数乘 | Multiply each component 每个分量相乘 | 3(2, −1) = (6, −3) |
3. Position Vectors and Displacement Vectors | 位置向量与位移向量
If O is the origin, the position vector of point A is OA = a. The displacement from A to B is AB = b − a. This relation is the foundation of most geometric proofs in B9i Part 3.
如果 O 是原点，点 A 的位置向量为 OA = a。从 A 到 B 的位移向量为 AB = b − a。这个关系是 B9i 第三部分大多数几何证明的基础。
AB = OB − OA = b − a
Many mistakes come from writing AB = a − b. Always subtract the starting point’s position vector from the end point’s position vector.
许多错误来自于把 AB 写成 a − b。始终要用终点的位置向量减去起点的位置向量。
| Vector 向量 | Expression 表达式 |
| Position of A 点 A 的位置 | OA = a |
| Position of B 点 B 的位置 | OB = b |
| Displacement from A to B 从 A 到 B 的位移 | AB = b − a |
4. Parallel Vectors and Scalar Multiples | 平行向量与数乘关系
Two non-zero vectors are parallel exactly when one is a scalar multiple of the other. For example, if PQ = 3a and RS = 6a, then PQ and RS are parallel because RS = 2PQ.
两个非零向量平行当且仅当其中一个是另一个的数乘。例如，若 PQ = 3a，RS = 6a，则 PQ 与 RS 平行，因为 RS = 2PQ。
RS = 2PQ ⇒ RS ∥ PQ
This also works for unit vectors and column vectors. If AB = 4i + 2j and CD = 6i + 3j, then CD = 1.5(4i + 2j), so AB and CD are parallel.
这也适用于单位向量和列向量。若 AB = 4i + 2j，CD = 6i + 3j，则 CD = 1.5(4i + 2j)，因此 AB 与 CD 平行。
5. Collinearity and Geometric Proof | 共线关系与几何证明
Three points A, B and C are collinear if AB and AC are parallel and share a common point A. You can prove collinearity by showing that AC = kAB for some scalar k.
三点 A、B、C 共线，如果 AB 与 AC 平行且具有公共点 A。你可以通过证明 AC = kAB（k 为实数）来证明共线。
For instance, if AB = 2p + 4q and AC = 3p + 6q, then AC = 1.5(2p + 4q) = 1.5AB, so A, B and C are collinear because both vectors pass through A.
例如，若 AB = 2p + 4q，AC = 3p + 6q，则 AC = 1.5(2p + 4q) = 1.5AB，因此 A、B、C 共线，因为两个向量都经过点 A。
Always state the common point explicitly. Writing only AC = 1.5AB is not enough unless you say that A is shared by both vectors.
务必明确指出公共点。只写 AC = 1.5AB 还不够，必须说明 A 是这两个向量的公共点。
6. Midpoints and Section Formula in Vector Form | 中点与向量分点公式
If M is the midpoint of AB, its position vector is m = (a + b)/2. More generally, if a point P divides AB in the ratio m : n, then p = (na + mb)/(m + n).
如果 M 是 AB 的中点，则其位置向量为 m = (a + b)/2。更一般地，若点 P 按照 m : n 划分 AB，则 p = (na + mb)/(m + n)。
m = (a + b) ÷ 2
p = (na + mb) ÷ (m + n)
For example, if P divides AB in the ratio 2 : 3, then p = (3a + 2b) ÷ 5. This is useful in triangle and parallelogram proofs where a point is not always exactly at the midpoint.
例如，若点 P 按照 2 : 3 划分 AB，则 p = (3a + 2b) ÷ 5。这在三角形和平行四边形证明中非常有用，因为点不一定总是恰好在中点处。
7. Proving Properties of Triangles and Parallelograms | 三角形与平行四边形性质的向量证明
With vectors you can prove geometric facts such as: the midpoints of the sides of a triangle form a parallelogram, or the diagonals of a parallelogram bisect each other. For example, if ABCD is a parallelogram, then AB = DC and AD = BC.
借助向量可以证明几何事实，例如：三角形各边中点构成平行四边形，或平行四边形的对角线互相平分。例如，若 ABCD 是平行四边形，则 AB = DC 且 AD = BC。
To prove that the diagonals of a parallelogram bisect each other, let the position vectors of A, B and D be a, b and d. Then C = b + d − a. The midpoint of AC is (a
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